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For a direct sum this is clear, as one can inject from or project to the summands. For a left split sequence, the map t × r: B → A × C gives an isomorphism, so B is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection C → A × C gives an injection C → B splitting r (2.).
In applied mathematics and computer science, variable splitting is a decomposition method that relaxes a set of constraints. [1] Details. When the variable ...
The splitting field of x q − x over F p is the unique finite field F q for q = p n. [2] Sometimes this field is denoted by GF(q). The splitting field of x 2 + 1 over F 7 is F 49; the polynomial has no roots in F 7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3]
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices.
Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z. The categorical concept of a section is important in homological algebra , and is also closely related to the notion of a section of a fiber bundle in topology : in the latter case, a section of a fiber bundle is a section ...
Horizontal partitioning is a database design principle whereby rows of a database table are held separately, rather than being split into columns (which is what normalization and vertical partitioning do, to differing extents). Each partition forms part of a shard, which may in turn be located on a separate database server or physical location.
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product).
The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject.